In this talk, we present numerical simulations of the nonlinear
evolution of a forced wave packet in the presence of a critical layer
in a shear flow. Two different geophysical flows are considered:
vertically propagating internal gravity wave packets in a stratified
shear flow and Rossby wave packets propagating toward the equator in a
zonal flow. It is believed that critical layer dynamics are pertinent
to phenomena observed in the atmosphere and ocean, such as
stratospheric sudden warmings in the northern hemisphere and the
quasi-biennial oscillation. The theories associated with gravity wave
critical layers may explain many aspects of the dynamics of waves
forced by flow over topography, e.g., momentum transport, wave
breaking and the generation of turbulence.

Most previous analyses of critical layer phenomena have dealt with
spatially periodic, monochromatic waves. For a more realistic
representation of wave activities in the atmosphere, we employ a
forcing in the form of a spatially localized wave packet, rather than a
monochromatic wave. We solve the nonlinear equations numerically using
a pseudo-spectral Fourier approximation and a high order compact finite
difference scheme. It is found that the spatial localization delays the
onset of the nonlinear breakdown in the critical layer. The
disturbance is absorbed at the critical layer and, unlike in the
monochromatic case, this continues for large time because the
convergence of momentum flux into the critical layer is balanced by an
outward flux of momentum in the horizontal or zonal direction. In the
gravity wave packet problem, we observe also that the prolonged
absorption of the disturbance stabilizes the solution to the extent
that it is always convectively stable; the local Richardson number
remains positive well into the nonlinear regime.

A three-dimensional finite element model is used to simulate summer
currents for Juan de Fuca Strait and the southwestern continental
margin of Vancouver Island. The calculations are forced and/or
initialized with seasonal winds, climatological density fields, and
elevation-specified boundary conditions that have been adjusted via
inversion to more accurately represent the California Undercurrent.
Tides are also included to correctly represent turbulent mixing, bottom
friction, and the contribution of tidal rectification. The seasonal
model currents are shown to compare favourably with multi-year,
low-pass filtered current meter observations and to capture strong
shears both vertically in Juan de Fuca Strait, and horizontally and
vertically across the continental shelf and slope. The tidal currents
are demonstrated to be more accurate than those computed by a
three-dimensional model with homogeneous density.

McDougall and Garrett have pointed out that there is an error of as
much as 30% in the averaged tracer equations currently carried by
Boussinesq ocean models. This is in addition to the error of about
5% usually associated with the Boussinesq approximation. New
satellite mounted gravity measuring instruments will measure bottom
pressure changes to within the accuracy of the Boussinesq
approximation. In this talk, it is pointed out that by reinterpreting
the model variables most of the 30% error is eliminated. Further,
with very little extra cost in terms of cpu requirement, models can
easily be made fully non-Boussinesq. Such models not only have the
advantage that the averaged tracer equation is now accurately
represented, but also conserve mass rather than volume and so
accurately compute changes in bottom pressure and sea level. This is
important for comparing model-computed sea level and bottom pressure
with observations, and also for computing changes in sea level
associated with climate change.

Inspired by laboratory experiments and eddy-resolving numerical
simulations, we present a detailed theory of the structure of a
circumpolar current. Using data from laboratory and numerical
experiments, we derive empirical relationships for the depth of
penetration, width, and strength of the current in terms of the
surface buoyancy fluxes and wind stress.

In our laboratory and numerical experiments, a circumpolar current in
response to a temperature gradient formed between cooled polar waters
and warmed tropical waters. The current is enhanced by a wind
stress which concentrates and pumps the warm, tropical surface water
to depth while up-welling the deep, cold polar waters. Combined, the
forcing acts to overturn the isotherms resulting in a strong horizontal
temperature front. This front is baroclinically unstable, and growing
waves feed off the available potential energy forming vigorous eddies.
The resulting eddies play an essential role in transporting heat across
the current by sweeping warm tropical water into the polar region where
it is cooled, smoothing out the temperature front. When the heat flux
across the front produced by the eddies balances the heat flux produced
by the surface forcing, a statistically steady current is achieved.

In order to develop a simple theory of the circumpolar current, we
examine a steady-state, two dimensional model in which the
eddy-transport has been parameterized in terms of the mean buoyancy
gradient and thermal wind. We are able to establish the general
structure of the steady current from buoyancy budgets considerations.
Using our experimental data to empirically determine the remaining
constants of proortionality, we derive straight forward formulas that
relate the depth of penetration, the width, and the speed of the
current to the surface buoyancy flux and wind stress. Despite the
simplicity of our assumptions, these formula give surprisingly
reasonable predictions of the observed ACC when forced with observed
surface fluxes.

Mesoscale eddies (50-100 km scales) play an important role in
determining the overall circulation of the world oceans and are
intimately related to the transport of variables like heat, vorticity
and momentum. Since a proper resolution of these eddy fields in long
term simulations is computationally prohibitive, it is important to be
able to parameterize their effects on mean flow. In this talk, we will
consider two idealized setups to study such eddy fields. First, we
will briefly present simulations of a bouyancy driven,
baroclinically-unstable zonal flow in a channel and discuss the salient
features of the associated meridional transport. Next, and in more
detail, we will consider the case of a double-gyre wind forced
barotropically-unstable flow in a rectangular midlatitude ocean basin.
In this latter case-a much studied classical problem-we show that a
new component of time-mean flow, which is eddy-driven, arises when
dissipation is low enough. Finally, we present models for such a flow
in which eddy effects are parameterized. One of these uses nonlinear
dispersion to model the effects of eddies on mean flow and is based on
the ideal mean flow fluid model of Holm, Marsden, and Raitu (PRL 80,
4273-4277, 1998).

The logarithmic law of the wall is still considered one of the main
building blocks in turbulence theory. It has been verified in innumerable
experiments and has been implemented in almost all statistical turbulence
models. Using group theoretical methods it has been shown that the law of the
wall and a variety of new turbulence laws can be derived from first principles
solely using Navier-Stokes equations.

The characteristic secondary flow pattern, so called Dean vortices,
that develops in laminar flow through pipe bends is well known and
equally well understood. The turbulent flow through pipe bends,
however, is more complicated, and only a few numerical simulations were
carried out to investigate this flow. An interesting flow phenomenon in
a 90°-bend is the so called swirl switching which, to the
author's knowledge, has not yet been studied using numerical methods.
>From an experimental investigation a tangential flow in alternating
direction with a low frequency at the inner side of the curved pipe is
reported. This paper investigates numerically such low frequency
phenomena in pipe bends.

Large-eddy simulations are performed using an AUSM method with central
differencing for the pressure term and a Runge-Kutta time stepping
scheme. Results will be presented for two 90°-bend radii at
three Reynolds numbers. Time-averaged and statistical data will be
compared with experimental findings.

We consider the growth of 3 dimensional perturbations to the
hydrostatic equations linearized about a chaotic 2 dimensional base
state. We first consider the case of a nonrotating, homogeneous layer
of fluid. Based on an argument by Warn (personal communication), we
show numerically that the 3D perturbations grow exponentially. We
further argue that a secondary, nonhydrostatic instability occurs and
acts to curb the growth. This leads to an estimate of the saturation
level of the 3D perturbations. Next, the effect of rotation on the
primary instability mechanism is considered. Preliminary results
suggest that significant growth does not occur unless the maximum
Rossby number is greater than unity. Since vorticity distributions in
real turbulent flows show very non-Gaussian distributions (i.e.,
wide tails), this suggests that that instability occurs at r.m.s.
Rossby numbers that are relatively small. Finally, we consider how
stratification affects the initial growth of the 3D perturbations.
The results have implications to the transfer of energy from
``balanced'' to unbalanced (or forward cascading) modes and, by
extension, to the mechanisms whereby energy is dissipated from the
general circulation of the oceans and atmosphere.